Functiones et Approximatio Commentarii Mathematici

Representation of a rational number as a sum of ninth or higher odd powers

M.A Reynya

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In the present paper, we substantially generalize one of the results obtained in our earlier paper [RM]. We present a solution of a problem of Waring type: if $F(x_1, \dots ,x_N)$ is a~symmetric form of odd degree $n\ge 9$ in $N=16\cdot 2^{n-9}$ variables, then for any $q\in \mathbb{Q}$, $q\neq 0$, the equation $F(x_i)=q$ has rational parametric solutions, that depend on $n-8$ parameters.

Article information

Source
Funct. Approx. Comment. Math., Volume 58, Number 1 (2018), 79-87.

Dates
First available in Project Euclid: 5 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.facm/1493949627

Digital Object Identifier
doi:10.7169/facm/1646

Mathematical Reviews number (MathSciNet)
MR3780035

Zentralblatt MATH identifier
06924917

Subjects
Primary: 11D72: Equations in many variables [See also 11P55]
Secondary: 11P05: Waring's problem and variants

Keywords
Diophantine equations parametric solutions

Citation

Reynya, M.A. Representation of a rational number as a sum of ninth or higher odd powers. Funct. Approx. Comment. Math. 58 (2018), no. 1, 79--87. doi:10.7169/facm/1646. https://projecteuclid.org/euclid.facm/1493949627


Export citation

References

  • A. Bremner, Geometric approach to equal sums of fifth powers, J. Number Theory 13 (1981) 337–354.
  • A. Choudhry, On equal sums of fifth powers, Indian J. Pure Appl. Math. 28 (1997) 1443–1450.
  • A. Choudhry, On sums of seventh powers, J. Number Theory 4 (2000) 266–269.
  • R.L. Ekl, New results in equal sums of like powers, Math. Comput. 67 (1998) 1309–1315.
  • L.J. Lander and T.R. Parkin, A counterexample to Euler's sum of powers conjecture, Math. Comput. 21 (1967) 101–103.
  • L.J. Lander, T.R. Parkin, and J.L. Selfridge, A survey of equal sums of like powers, Math. Comput. 21 (1967) 446–459.
  • K.S. Rao, On sums of fifth powers, J. London Math. Soc. 9 (1934) 170–171.
  • M.A. Reynya, Symmetric homogeneous Diophantine equations of odd degree, Intern. J. Number Theory 28 (2013) 867–879.
  • H.P.F. Swinnerton-Dyer, A solution of $A^5+B^5+C^5=D^5+E^5+F^5$, Proc. Cambridge. Phil. Soc. 48 (1952) 516–518.
  • B.L. van der Waerden, Algebra, volume 1, transl. from the seventh German edition, Springer-Verlag, New York, 1991.
  • L.E. Dickson, Terry-Escott Problem, in: L.E. Dickson, History of the theory of numbers, Vol. 2, Chelsea 1992, reprint 28 (1992) 710–711.